Question 3:Prove that 1 x 3 + 2 x 5+ 3 x 7+...+ n x (2n+1)=n(n+1)(4n+5)/6 for all natural numbers n.

Question 4:Prove that 1/(1 x 2) + 1/(2 x 3) + 1/(3 x 4) +...+1/(n x (n+1)=1-(1/(n+1)) for all natural numbers n.

1)

The general term is

n(2n+1)= 2n^{2}+n

Sum of squares of first n natural numbers is n(n+1)(2n+1)/6

Sum of first n natural numbers is n(n+1)/2

Hence 2 * n(n+1)(2n+1)/6 + n(n+1)/2

= n(n+1)/6 * [ 2(2n+1)+3]= n(n+1)/6 *[4n+5]

2)

1/(1 x 2)= 1-1/2

1/(2 x 3 ) = 1/2 -1/3

1/(3 x 4) = 1/3 -1/4

…..

1/n(n+1) = 1/n -1/(n+1)

Adding all the equations

= 1 – 1/(n+1)

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